Mathematical Modeling in Combustion Science: Proceedings of a Conference Held in Juneau, Alaska, August 17–21, 1987

166 Pages·1988·7.47 MB·English

by  A. K. Kapila (auth.)

.i High Mach Number Combustion

NOITAITINI-KCOHS FO A ENALP NOITANOTED EVAW
A. K. Kapila
tnemtrapeD of Mathematical Sciences, Rensselaer Polytechnic Institute
Troy, weN York 0953-08121
Introduction.
This paper gives a mathematical description of the early stages of evolution
of a planar detonation wave, initiated yb the egassap of a strong shock. It is
demussa that the reactive sag seogrednu a one-step, first-order, irreversible -ed
composition reaction governed yb Arrhenius kinetics. ehT analysis is asymptotic,
in the limit of large activation energy. Theirse a deliberate attempt at brevity,
since the following presentation draws heavily the upon study reported in [1], to
which the reader is referred for further details.
ehT basis configuration is sa follows. roF time t < ,O the half ecaps x > 0
is filled with a reactive sag at a uniform state of rest, dna at temperature low
hguone for the chemical reaction rate to eb negligible over yna time scales of
interest. At t = ,+0 a piston initially at x = 0 is pushed into the sag at a con-
stant speed, thereby generating a shock evaw running daeha of it. If the sag erew
inert, the shock would propagate steadily dna maintain a fixed strength. It is
,demussa however, that the shock switches no a significant tnuoma of chemical acti-
vity in the sag behind it, which then sah the effect of strengthening dna accel-
erating the shock. ehT mia of the following analysis is to describe the ecneuqes
of events in the dekcohs sag until a detonation is about to form.
Governing Equations.
ehT relevant equations are the reactive Euler equations, which for planar,
one-dimensional motion are:
(1) tP + u xP + p xU = ,O
(2) tUCP + )xUU + 1 xP = ,O
(3) piT t + u Txl - y-1 Ip t + u xp = ,w~
Y
(4) plY t + u xY I = - ,w
(5) p = ,Tp
(6) w = i yp pxe 0 - 0.
Here p, p, ,T u, dna Y are, respectively, the sag pressure, density, temperature,
velocity dna reactant ssam fraction. ehT reference coordinate frame sah neeb
selected to evom with the piston face, dna the reference state of the sag is taken
to eb the shocked state at t = .+0 Velocity is referred to the acoustic speed dna
time to the induction time at the reference state; their product then defines the
reference length. ehT dimensionless parameters appearing above are the dimension-
less chemical heat release ,8 the specific heats ratio y dna the dimensionless
activation temperature o.
ehT appropriate boundary conditions for the shocked region under study are
u = 0 at x = ,O
dna the Rankine-Hugoniot jump conditions for ,T p, u dna p at the shock locus
Xs(t). Immediately behind the shock, Y sah the value unity.
In the following analysis, ~ dna y are demussa fixed dna 0(1), dna the -pmysa
totic limit 0 ÷ ~ is employed. ehT resulting sequence of events proceeds sa
fol lows:
Induction State
Initially, T-I = OIe-zl is na appropriate range to consider, sa the reaction-
rate expression in (6) suggests. Accordingly, eno sets
u = 8-1u I + ...; ~ = 1 + 0-i~ I + ... for ~ = ,T p, p dna ,Y
to obtain leading-order reduced equations
(7) (--~ ±--~)Pz ± Y uz = Y expCTz'
at x5
(8) _Aa t (TI -~ - pl : expCTI~'
(9) lP = lP - TI' zY~ = _ i
aT ~ "~1TCpxe
esehT equations are subject to
ul(O,t) = 0
dna the linearized H-R conditions at
x = Mt,
the undisturbed shock locus, where M is the initial shock speed. ehT above pro-
blem can eb solved numerically, first for TI, Pl dna u I dna then for Pl dna YI
(see, e.g., 2 or 3 for details). ehT solution exhibits thermal runaway, which
is characterized by the appearance of logarithmic singularities in T I dna zP (and
therefore YI I, first at the piston face, at a definite time te(M). It is note-
worthy that Pl remains bounded. Typical profiles of T1(x,t) for t smaller than,
but close to, t e are shown in Figure i. These profiles display the emergence of a
%
i~~,,~HOC K L S 0 U C
0
Figure I
shrinking boundary layer near x = 0 within which the solution grows rapidly. enO
can, in fact, continue the induction solution beyond t > te; the locus of infini-
ties in the solution then moves into the interior of the domain (Figure 2) at a
speed which is initially supersonic, but fails monotonically to the sonic value.
More about this locus will eb said later.
It is possible to give na analytical description of the boundary layer at the
piston face in the limit ~ + ,+O where
= te - t.
ehT boundary layer is found to eb OI~Y/(2y-1) ) thick, dna therefore, describable
in terms of the spatial coordinate ,~ defined yb
x = 5 ~y/(2y-1).
PATH OF INDUCT/ON
S I N G U L ~
----'~"11~...,...,._.. I~ O R, O
RAW
"~ / / ACT HARACTERISTIC ~URN7 ~TATE
Figure 2
ehT boundary-layer solution is found to have the form
T ~ - ~n(y~) + F (~) + ... ,
1 o
Pl ~ - zn(BlS) + F0(~) + ....
u I ~ ~ (Y-I)/(2y-I)~n~ )~(0H + HI(~) + ....
erehw
)E(oF = - n~ i + ~ A I K(2y-1)Iy ,
E(oH ) = _ y-1 ~ zA ~(y-1)/y ,
2y
HI(K ) = 2y-i 2y ~ AI ~(y-1)/y n~ ~ + Fo(~) + 2B ,
dna the constants ~, A I, B dna B 2 are known. Observe that to leading order the
boundary layer displays alspatially uniform growth of temperature dna pressure,
with spatial structure appearing only sa a perturbation. ehT structure is singu
lar at ~ = ,O but this singularity nac eb devomer yb snaem of a thinner, inner
layer in which x = 0(~); details nac eb found in I..
Explosion egatS
ehT layer solution semoceb nonuniform nehw -zn(~) semoceb 0(8), suggesting
that further evolution should occur no the time scale ,~ defined yb
~=e , ~>0 .
The solution now turns out to have the expansions
(i0) T ~ T0(o) + e -i Ti(K,~) + ....
(Ii) p ~ T )~( + ~)-i
o Pl (E'°) + ....
)21( u = 0e -O~(Y-1)/(2Y-1)
(13) Y ~ Y0(o) + e -I YI(E,~) + ....
(14) p 1 + 8 -I +
~ 1% " " "
erehw
(15) To(~ ) = 1 yo(~ ) = I+~y-To(~)
' x~ '
)61( ~zT = - 02T zny Yo/T~ - T o2 nz i + 1A~ --TT ~ (2Y-I)IY, zP = TI + oT zoP ,
o
)71( Yl = - i T + B 3
C--B i
dna dna B are nwonk constants. Observe that density is essentially
10P 3
degnahcnu from its value at runaway, i.e., the material within the layer is
inertially confined. sA ~ increases, T dna p increase saerehw Y decreases.
Eventually, p dna T kaep nehw 'oY the leading term in ,Y vanishes. This sneppah
at
= ~yl(l+By) ,
dna the peak values era
T ~ ,yB+1 p ~ l+~y .
At the emas time, the 0C0-II term (in T, say; ees (16)) develops a logarithmic
singularity, indicating the nwodkaerb of the solution.
Before advancing further in time, it is instructive to point out that sa the
boundary layer recedes towards the piston during the explosion stage, it leaves
behind it na exponentially thin intermediate region in which the solution is
essentially stationary in time, but is not close to the induction solution. This
region is governed yb the spatial variable ,X defined yb
x = o_e x
, 0 < X < ~y/(2y-1)
where the left restriction no X corresponds to the edge of the boundary layer dna
T"
i
I
!
J io
FIq-
L A Y~/:R INTEIRI',/IEZ31AT£ REGION
Figure 3
the right restriction to merging with the outer region (see Figure 3). ehT sol u-
tion in the intermediate region is given yb
I
T ~.
t _ 2y-\ x
Y
with analogous expressions for p dna ,Y while
p ~ i , u ~ ..1..-y (2y-1)c XIA .
3y
Transition stage
Further evolution of the solution near the piston face occurs no the time
scale ~, defined yb
t = t + v(e)e
e
erehw
v = BeexpE-6ye/{t+#y} •
The corresponding spatial variable in the boundary layer (in view of its
O(~y/(2Y -1)) thickness), is z, defined yb
x : vY/(2Y-l)z •
ehT transition zone analysis is rather involved, but the asymptotic form of its
solution for large @ dna z is particularly simple dna of special interest. enO
finds that
T ~ l+~y - e -I T l
in this zone, dna that
Y
~ Xlexp-¢ + A z (2Y-I)/Y + constant sa ,¢ z ÷ ~ ,
erehw X dna A are nwonk constants. Thus, a reaction evaw is born, propagating out
of the transition zone with velocity
dx/dt ~ v-(Y-1)/(2y-1)dz/d¢ ~ x -(Y-I)/Y •
Behind the evaw the sag is completely reacted, dna pressure dna temperature era at
kaep values, l+~y to leading order, while density sah eht leading-order value of
unity dna velocity is exponentially small. sA time continues to evolve, this evaw
speews across the intermediate zone via a ecneuqes of vonemeS explosions. Parti-
cle velocities, dna ecneh density ,segnahc continue to eb exponentially small
during this process; the evaw is completely reactive in character. In fact, it
nac eb thought of sa a nearly-constant-density detonation wave, travelling at
velocity supersonic relative to the burnt sag behind it.
sA the evaw propagates across the intermediate zone, its velocity sesaerced
esuaceb of the falling temperature immediately ahead of it, dna yb the time eht
evaw reaches the edge of the intermediate zone, its velocity sah fallen to within
a wef multiples of the dnuos deeps in the burnt gas. Local chemical times evah
won risen sufficiently to eb elbarapmoc to the local acoustic times, dna the sag
in the vicinity of the evaw is on longer inertially confined. ehT evaw path tsum
won eb detupmoc numerically, dna is in fact given to a first approximation yb the
thermal yawanur locus detupmoc yb the induction zone analysis (Figure 2). nevE
the numerical description is valid only os long sa the evaw remains supersonic,
breaking nwod at point S in Figure 2 where the detupmoc evaw path is tangential to
the forward characteristic originating in the burnt gas. At this point, kcohs
formation, dna eht tneuqesnoc birth of a conventional detonation, is imminent. A
full description of these sessecorp will eb given elsewhere.
01
stne.megdelwonkcA
This paper sah benefited from discussions with J. .W Dold. ehT research re-
ported here saw supported yb the .U .S ymrA hcraeseR Office dna yb the soL M soma
National Laboratories.
secnerefeR
1. T. L. Jackson, .A .K Kapila dna .D .S Stewart, Evolution of a reaction
center in na explosive material, MAIS J. Appl. Math. Submitted for
publication (1987).
2. T. L. Jackson dna .A .K Kapila, Shock-induced thermal runaway, MAIS J.
Appl. Math., ,54 031 (1985).
3. J. .F Clarke dna .R .S Cant, ydaetsnoN cimanydsag effects in the induction
niamod behind a strong shock wave, Progress in Astro. dna Aero., ,59 241
(1984).